nt.number theory – Simultaneous reductions of elliptic curves: same number of points $|E(Bbb F_p)| = |E'(Bbb F_p)|$ for some prime $p$?


Let $E,E’$ be two elliptic curves over $Q$. Is there at least one prime number $p geq 5$ of good reduction for $E,E’$ such that $|E(F_p)| = |E'(F_p)|$ ?

If $E,E’$ are isogenous over $Q$, this is clearly true. In any case, Fran├žois Charles proved that there are infinitely many primes $p$ such that the reductions of $E,E’$ mod $p$ are isogenous over $overline{F_p}$. If the result holds over $F_p$, we are done.

For instance, for the curves $y^2 = x^{3} – 3 x – 3, y^2 = x^{3} + x + 6$, the smallest such prime is $p = 3121$.

One could ask if the above question can be generalized to arbitrary smooth projective irreducible curves $C,C’$ (not necessarily of the same genus!) over $Q$, or even over a number field $K$ (though the heuristic only works well if the primes $mathfrak{p}$ are totally split over $Q$, or at least have degree $f leq 2$).

Using ideas from there, one can show that if $E,E’$ are not isogenous over $overline{Q}$, then the set of such primes has density $0$. According to Lang–Trotter heuristics, one should expect the number of such primes $p<X$ to be $sim c sqrt{x} / log(x)$ for some $c geq 0$ (if $c neq 0$, this should give infinitely many such primes).

Note: in his talk, he mentions that he doesn’t know what happens for three elliptic curves, nor what happens over $F_p$ instead of $overline{F_p}$.
For four (or more) elliptic curves, one should not expect infinitely many primes such that $|E_i(F_p)|$ are all equal when $1 leq i leq 4$.